AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 291397 10.1155/2014/291397 291397 Research Article Boundedness of One-Sided Oscillatory Integral Operators on Weighted Lebesgue Spaces http://orcid.org/0000-0001-9109-4142 Fu Zunwei 1 http://orcid.org/0000-0002-3259-6684 Lu Shanzhen 2 Pan Yibiao 3 Shi Shaoguang 1 Mohiuddine S. A. 1 Department of Mathematics Linyi University, Linyi 276005 China lyu.edu.cn 2 School of Mathematical Sciences Beijing Normal University Beijing 100875 China bnu.edu.cn 3 Department of Mathematics University of Pittsburgh Pittsburgh, PA 15260 USA pitt.edu 2014 322014 2014 06 10 2013 09 12 2013 3 2 2014 2014 Copyright © 2014 Zunwei Fu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We consider one-sided weight classes of Muckenhoupt type, but larger than the classical Muckenhoupt classes, and study the boundedness of one-sided oscillatory integral operators on weighted Lebesgue spaces using interpolation of operators with change of measures.

1. Introduction and Main Results

Oscillatory integrals in one form or another have been an essential part of harmonic analysis from the very beginnings of that subject; three chapters are devoted to them in the celebrated Stein's book . Many operators in harmonic analysis or partial differential equations are related to some versions of oscillatory integrals, such as the Fourier transform, the Bochner-Riesz means, and the Radon transform which has important applications in the CT technology. Among numerous papers dealing with oscillatory singular integral operators in some function spaces, we refer to  and the references therein. More generally, let us now consider a class of oscillatory integrals defined by Ricci and Stein : (1)Tf(x)=p.v.eiP(x,y)K(x-y)f(y)dy, where P(x,y) is a real-valued polynomial defined on × and the function KC1({0}) is a Calderón-Zygmund kernel. That means K satisfies (2)|K(x)|C|x|,|K(x)|C|x|2,x0,(3)a<|x|<bK(x)dx=0a,b(0<a<b). Throughout this paper, the letter C will denote a positive constant which may vary from line to line but will remain independent of the relevant quantities.

We state a celebrated result of Ricci and Stein on oscillatory integrals as follows.

Theorem 1 (see [<xref ref-type="bibr" rid="B16">8</xref>]).

Let 1<p<, K satisfy (2) and (3). Then for any real-valued polynomial P(x,y), the oscillatory integral operator T is of type (Lp,Lp) and its norm depends on the total degree of P, but not on the coefficients of P in other respects.

Weighted inequalities arise naturally in harmonic analysis, but their use is best justified by the variety of applications in which they appear. It is worth pointing out that many authors are interested in the inequalities when the weight functions belong to the Muckenhoupt classes (), which are denoted by Ap(1<p<) classes for simplicity. This class consists of positive locally integrable functions (weight functions) w for which (4)supI(1|I|Iw(x)dx)(1|I|Iw(x)1-pdx)p-1<, where the supremum is taken over all intervals I and 1/p+1/p=1.

In 1992, Lu and Zhang  established the weighted version of Theorem 1.

Theorem 2.

Let p,P(x,y) and K be as in Theorem 1. Then the oscillatory singular integral operator T is of type (Lp(w), Lp(w)) with wAp. Here its operator norm is bounded by a constant depending on the total degree of P, but not on the coefficients of P in other respects.

We point out that Theorems 1 and 2 also hold for dimension n2. We choose the results for n=1 here in order to introduce the one-sided operators which were defined on . Theorems 1 and 2 are also true for more general kernels, that is, nonconvolution kernels, under the L2-boundedness assumption on the corresponding Calderón-Zygmund singular integral operators: (5)T~(x)=p.v.nK(x-y)f(y)dy. However, this topic exceeds the scope of this paper. For more information about this work, see [8, 10], for example.

The study of weights for one-sided operators was motivated not only by the generalization of the theory of both-sided ones, but also by their natural appearance in harmonic analysis; for example, they are required when we treat the one-sided Hardy-Littlewood maximal operator : (6)M+f(x)=suph>01hxx+h|f(y)|dy,M-f(x)=suph>01hx-hx|f(y)|dy, arising in the ergodic maximal function. Sawyer first introduced the classical one-sided weight Ap+ classes in . The general definitions of Ap+ and Ap- were introduced in  as (7)Ap+:supa<b<c1(c-a)pabw(x)dx(bcw(x)1-pdx)p-1C,Ap-:supa<b<c1(c-a)pbcw(x)dx(abw(x)1-pdx)p-1C, where 1<p<, 1/p+1/p=1; also, for p=1, (8)A1+:M-wCw,A1-:M+wCw.

The smallest constant C for which the above inequalities are satisfied will be denoted by Ap+(w) and Ap-(w), p1. Ap+(w) (Ap-(w)) will be called the Ap+ (resp., Ap-) constant of w. By Lebesgue's differentiation theorem, we can easily prove A1+(w) (resp., A1-(w))1. In , the class A+ was introduced as A+=p<Ap+ (see also ). It is easy to see that for 1p, ApAp+, ApAp-, and Ap=Ap+Ap-.

Theorem 3 (see [<xref ref-type="bibr" rid="B18">11</xref>]).

Let 1<p<. Then

M+ is bounded in Lp(w) if and only if wAp+;

M- is bounded in Lp(w) if and only if wAp-.

The one-sided weight classes are of interest, not only because they control the boundedness of the one-sided Hardy-Littlewood maximal operator, but also because they are the right classes for the weighted estimates of one-sided Calderón-Zygmund singular integral operators , which are defined by (9)T~+f(x)=limε0+x+εK(x-y)f(y)dy,T~-f(x)=limε0+-x-εK(x-y)f(y)dy, where K is the one-sided Calderón-Zygmund kernel with support in -=(-,0) and +=(0,+), respectively. We say a function K is a one-sided Calderón-Zygmund kernel if K satisfies (2) and (10)|a<|x|<bK(x)dx|C,0<a<b with support in -=(-,0) or +=(0,+). An example of such a kernel is (11)K(x)=sin(log|x|)(xlog|x|)χ(-,0)(x), where χE denotes the characteristic function of a set E.

Theorem 4 (see [<xref ref-type="bibr" rid="B1">15</xref>]).

Let 1<p< and K be a one-sided Calderón-Zygmund kernel. Then

T~+ is bounded in Lp(w) if and only if wAp+;

T~- is bounded in Lp(w) if and only if wAp-.

Theorem 4 is the one-sided version of weighted norm inequality of singular integral due to Coifman and Fefferman .

Highly inspired by the above statements for oscillatory singular integral operators and one-sided operator theory, in , the authors had introduced the one-sided oscillatory singular integral operators and studied the weighted weak type (1,1) norm inequalities for these operators. In this paper, we will further study the one-sided Muckenhoupt weight classes and give the one-sided version of Theorem 2. It is well known that the property of the one-sided Muckenhoupt weight classes is worse than the Muckenhoupt weight classes (see also ). For example, both the reverse Hölder inequality and the doubling condition are not true for the one-sided case. Therefore, some new methods are needed to deal with some new difficulties.

We first recall the definition of one-sided oscillatory integral operator as (12)T+f(x)=limε0+x+εeiP(x,y)K(x-y)f(y)dy,T-f(x)=limε0+-x-εeiP(x,y)K(x-y)f(y)dy, where P(x,y) is a real-valued polynomial defined on × and the kernel K is a one-sided Calderón-Zygmund kernel with support in - and +, respectively. Now, we formulate our results as follows.

Theorem 5.

Let 1<p< and K be a one-sided Calderón-Zygmund kernel. Then for any real-valued polynomial P(x,y),

there exists constant C>0 such that (13)T+fLp(w)CfLp(w),

where wAp+ and the operator norm depend on the total degree of P and Ap+(w), but not on the coefficients of P in other respects;

there exists constant C>0 such that (14)T-fLp(w)CfLp(w),

where wAp- and the operator norm depend on the total degree of P and Ap-(w), but not on the coefficients of P in other respects.

The rest of this paper is devoted to the argument for Theorem 5. Section 2 contains some preliminaries which are essential to our proof. In Section 3, we will give the proof of Theorem 5.

2. Preliminaries Lemma 6 (see [<xref ref-type="bibr" rid="B18">11</xref>, <xref ref-type="bibr" rid="B17">18</xref>]).

Let 1<p< and w0 be locally integrable. Then the following statements are equivalent:

wAp+;

w1-pAp-;

there exist w1A1+ and w2A1- such that w=w1(w2)1-p.

According to the definition of Ap+, we can easily obtain the following lemma.

Lemma 7.

Let 1<p< and wAp+. Then Ap+(δλ(w))=Ap+(w), where δλ(w)(x)=w(λx) for all λ>0.

Proof.

For 1<p<, if wAp+, then (15)supa<b<c1(c-a)pabw(x)dx(bcw(x)1-pdx)p-1C. For λ>0, a=λa, b=λb, c=λc, and d=λd, we have (16)1(c-a)pabw(λx)dx(bcw(λx)1-pdx)p-1=1(c-a)paλbλw(x)λ-1dx(bλcλw(x)1-pλ-1dx)p-1=1(λ(c-a))paλbλw(x)dx(bλcλw(x)1-pdx)p-1=1(c-a)p  abw(x)dx(bcw(x)1-pdx)p-1C. The proof is complete.

We say a weight w satisfies the one-sided reverse Hölder RHr+ condition  if there exists C>0 such that for any a<b and 1<r<, (17)abw(x)rdxC(M(wχ(a,b))(b))r-1abw(x)dx, where M is the classical Hardy-Littlewood maximal operator. The smallest such constant will be called the RHr+ constant of w and will be denoted by RHr+(w). Corresponding to the classical reverse Hölder inequality, (17) is named the weak reverse Hölder inequality. For r=, we say a weight w satisfies the one-sided reverse Hölder RH+ condition if there exists C>0 such that w(x)Cm+w(x) for almost all x where m+ is the one-sided minimal operator defined as (18)m+f(x)=infh>01hxx+h|f|dy. The smallest such constant will be called the RH+ constant of w and will be denoted by RH+(w). It is clear that RH+(w)1. In , the authors give several characterizations of RHr+ where the constants C are not necessary the same.

Lemma 8.

Let a<b<c<d, 1<r<, and w0 be locally integrable. Then the following statements are equivalent:

abw(x)rdxC(M(wχ(a,b))(b))r-1abw(x)dx;

(1/(b-a))abw(x)rdxC((1/(c-b))bcw(x)dx)r with b-a=2(c-b);

(1/(b-a))abw(x)rdxC((1/(d-c))cdw(x)dx)r with b-a=d-b=2(d-c);

(1/(b-a))abw(x)rdxC((1/(c-b))bcw(x)dx)r with b-a=c-b;

(1/(b-a))abw(x)rdxC((1/(d-c))cdw(x)dx)r with b-a=d-c=γ(d-a), 0<γ1/2.

Lemma 9 (see [<xref ref-type="bibr" rid="B17">18</xref>]).

A weight wAp+ for p>1 if and only if there exist 0<γ<1/2 and a constant Cγ such that for b-a=d-c=γ(d-a) with a<b<c<d, the following inequality holds: (19)abw(x)dx(cdw(x)1-pdx)p-1Cγ(b-a)p.

Combining the results in [12, 15, 18, 19], we can deduce Lemma 10. In what follows, we will include its proof with slight modifications for the sake of completeness.

Lemma 10.

Let wAp+. Then there exists ε>0 such that w1+εAp+.

Proof.

By Lemma 6, we have w=w1w21-p with w1A1+ and w2A1-. For fixed interval I=(a,b), we next claim that w1RHr+ for all 1<r<C/(C-1) with C=max{A1+(w1),A1-(w1)}>1. In fact, we consider the truncation of w at height H defined by wH=min{w1,H} which also satisfies A1+ condition (with a constant CHC). Therefore, if λI=M(wHχI)(b) and Sλ={xI:wH(x)>λ}, then we have (20)SλwH(x)dxCHλ|Sλ|,λλI. Indeed, it is straightforward if Sλ=I since (21)wH(Sλ)=abwH(x)dxλI(b-a)CHλ|Sλ|. We now assume SλI and fix ε>0 and an open set O such that SλOI with |O|ε+|Sλ|. Let Oi=(c,d), which is connected. There are two cases; that is, ac<d<b and ac<d=b. In the first case, it is easy to check that d is not contained in Sλ. By the definition of Sλ, w1+, we have cdwH(x)dxCHλ(d-c), while the second case is handled as the case Sλ=I since cdwH(x)dxC(b-c). Thus wH(Oi)CHλ|Oi|. Adding up with i, we get (22)wH(Sλ)wH(O)CHλ|Oi|CHλ(ε+|Sλ|). Therefore, we obtain (20). For fixed θ>-1, multiply both sides of (20) by λθ and integrate from λI to infinity; we can obtain (23)1θ+1I(wHθ+2-λIθ+1)(x)dxCHθ+2IwHθ+2(x)dx. Now if r=θ+2<CH/(CH-1), then 1/(θ+1)-CH/(θ+2)>0, which implies (24)IwHr(x)dxCHλIr-1IwH(x)dx=CH(M(wHχI)(b))r-1IwH(x)dx.

The inequality CHC implies CH/(CH-1)C/(C-1). Therefore, if rC/(C-1), then we have (25)IwHr(x)dx=CH(M(wHχI)(b))r-1abwH(x)dx=C(M(w1χ(a,b))(b))r-1abwH(x)dx. Hence w1RHr+ by the monotone convergence theorem. Since w2A1-, we next claim that w21-pRH+. In fact, for any interval I=(a,b), we have (26)(1|I|Iw2(x)dx)1-p1|I|Iw2(x)1-pdx by Hölder's inequality and the A1- condition. For almost every xI-=(2a-b,a), we have (27)Cw21|I|Iw2(x)dx. Thus, (28)w2(x)1-pC(1|I|Iw2(x)dx)1-pC1|I|Iw2(x)1-pdxC1b-xxbw2(x)1-pdx, which implies our claim. Hence, (29)1|I|Iwr1|I|Iw1rsupI(w2-r(p-1))C(1I1I1w1)r(1I1I1w21-p)rC(infI1w1)r(supI1w21-p)rC(infw1)r(1I2I2w21-p)rC(1I2I2w)r, where I1=(b,2b-a) and I2=(2b-a,3b-2a). By Lemma 8, we obtain wRHr+. Hence, w1-pRHr- for all 1<r<C/(C-1) by Lemma 6.

Let us fix a<d and choose b,c such that b-a=d-c=(d-a)/4 (e.g., we choose b=(d+3a)/4, c=(3d+a)/4). Following from the five points a,b,(b+c)/2,c,d, we have four intervals, namely, (30)I1=(a,b),I2=(b,(b+c)2),I3=((b+c)2,c),I4=(c,d). By Lemma 8, we have (31)1|I1|Iwr(1|I4|I4wr(1-p))p-1(1|I2|I2w)r(1|I3|I3w(1-p))r(p-1)Cr. Thus, wrAp+ by Lemma 9. Choosing 0<ε=r-1<1/(C-1), then we complete the proof of the lemma.

To prove Theorem 5, we still need a celebrated interpolation theorem of operators with change of measures.

Lemma 11 (see [<xref ref-type="bibr" rid="B19">20</xref>]).

Suppose that u0,v0,u1,v1 are positive weight functions and 1<p0, p1<. Assume sublinear operator S satisfies (32)SfLp0(u0)C0fLp0(v0),SfLp1(u1)C1fLp1(v1). Then, (33)SfLp(u)CfLp(v) holds for any 0<θ<1 and 1/p=θ/p0+(1-θ)/p1, where u=u0pθ/p0u1p(1-θ)/p1, v=v0pθ/p0v1p(1-θ)/p1, and CC0θC11-θ.

Lemmas 10 and 11 are the main tools in proving Theorem 5.

3. Proof of Theorem <xref ref-type="statement" rid="thm1.5">5</xref>

In this section, we will prove Theorem 5 by induction, which is partly motivated by [8, 10]. We begin with the proof of (1). For any nonzero real polynomial P(x,y) in x and y, there are k,l,m0 such that (34)P(x,y)=aklxkyl+R(x,y) with akl0 and (35)R(x,y)=0α<k,0βmaαβxαyβ+0β<lakβxkyβ. We will write dx(P)=k and dy(P)=l. Below we will carry out the argument by using a double induction on k and l.

If dx(P)=0 and dy(P) is arbitrary, then P(x,y)=P(y) and T+f can be written as (36)T+f(x)=limε0+x+εK(x-y)g(y)dy, where g(y)=eiP(y)f(y). Therefore, the conclusion of Theorem 5 follows from Theorem 4.

Let k1 and assume that the conclusion of Theorem 5 holds for all P(x,y) with dx(P)k-1 and dy(P) arbitrary.

We will now prove that the conclusion of Theorem 5 holds for all P(x,y) with dx(P)=k and dy(P) arbitrary.

If dx(P)=k and dy(P)=0, then (37)P(x,y)=ak0xk+Q(x,y) with dx(Q)k-1. By taking the factor eiak0xk out of the integral sign, we see that this case follows from the above inductive hypothesis.

Suppose l1 and the desired bound holds when dx(P)=k and dy(P)l-1.

Now, let P(x,y) be a polynomial with dx(P)=k and dy(P)=l, as given in (34).

Case 1 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M294"><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>=</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math></inline-formula>).

Write (38)T+f(x)=x1+xeiP(x,y)K(x-y)f(y)dy+j=12j-1+x2j+xeiP(x,y)K(x-y)f(y)dy=:T0+f(x)+j=1Tj+f(x). Take any h, and write (39)P(x,y)=akl(x-h)k(y-h)l+R(x,y,h), where the polynomial R(x,y,h) satisfies the induction assumption and the coefficients of R(x,y,h) depend on h.

We consider first the estimates for T0+. It is easy to check that (40)T0+f(x)=x1+xei(R(x,y,h)+akl(y-h)k+l)K(x-y)f(y)dy+x1+x{eiP(x,y)-ei(R(x,y,h)+akl(y-h)k+l)}×K(x-y)f(y)dy=:T01+f(x)+T02+f(x). Now we split f into three parts as (41)f(y)=f(y)χ{|y-h|<1/2}(y)    +f(y)χ{1/2|y-h|<5/4}(y)+f(y)χ{|y-h|5/4}(y)=:f1(y)+f2(y)+f3(y).

Observe that if |x-h|<1/4, then (42)T01+f1(x)=x1+xei(R(x,y,h)+akl(y-h)k+l)K(x-y)f1(y)dy. Thus, it follows from the induction assumption that (43)|x-h|<1/4|T01+f1(x)|pw(x)dxC|y-h|<1/2|f(y)|pw(y)dy, where C is independent of h and the coefficients of P(x,y).

Notice that if |x-h|<1/4,1/2|y-h|<5/4, then y-x>1/4. Thus, (44)|T01+f2(x)|Cx+1/4x+1|K(x-y)f2(y)|dyCM+(f2)(x). So we have (45)|x-h|<1/4|T01+f2(x)|pw(x)dxC|y-h|<5/4|f(y)|pw(y)dy, where C is independent of h and the coefficients of P(x,y).

Again observe that if |x-h|<1/4 and |y-h|5/4, then y-x>1. Thus, (46)T01+f3(x)=0.

Combining (43), (45), and (46), we get (47)|x-h|<1/4|T01+f(x)|pw(x)dxC|y-h|<5/4|f(y)|pw(y)dy, where C is independent of h and the coefficients of P(x,y).

Evidently, if |x-h|<1/4 and 0<y-x<1, then (48)|eiP(x,y)-ei(R(x,y,h)+akl(y-h)k+l)|C|akl||x-y|=C(y-x). Therefore, when |x-h|<1/4, we have (49)|T02+f(x)|Cxx+1|f(y)|dxCM+(f(·)χB(h,5/4)(·))(x). It follows from Theorem 3 that (50)|x-h|<1/4|T02+f(x)|pw(x)dxC|y-h|<5/4|f(y)|pw(y)dy, where C is independent of h and the coefficients of P(x,y).

From (47) and (50), it follows that the inequality (51)|x-h|<1/4|T0+f(x)|pw(x)dxC|y-h|<5/4|f(y)|pw(y)dy holds uniformly in h+, which implies (52)T0+fLp(w)CfLp(w), where C is independent of the coefficients of P(x,y) and wAp+.

We proceed with the proof of Theorem 5 with the estimates for Tj+f. Because of the size condition (2), we observe that for j1(53)|Tj+f(x)|2j-1+x2j+x|f(y)||x-y|dyCM+(f)(x), where C is independent of j. By Lemma 10, we know that there exists ε>0 such that w1+εAp+. Thus we have (54)Tj+fLp(w1+ε)CfLp(w1+ε), where C is independent of j. We now only need to recall Lemma 3.7 in  to see that (55)Tj+fLpC2-jδfLp, where C depends only on the total degree of P(x,y) and δ>0. It follows from (54), (55), and Lemma 11 that (56)Tj+fLp(w)C2-jθδfLp(w), where 0<θ<1, θ is independent of j, and C depends only on the total degree of P(x,y).

From (52) and (56), it is clear that when wAp+, (57)T+fLp(w)CfLp(w), where C depends only on the total degree of P(x,y).

Case 2 (<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M365"><mml:mo>|</mml:mo><mml:msub><mml:mrow><mml:mi>a</mml:mi></mml:mrow><mml:mrow><mml:mi>k</mml:mi><mml:mi>l</mml:mi></mml:mrow></mml:msub><mml:mo>|</mml:mo><mml:mo>≠</mml:mo><mml:mn mathvariant="normal">1</mml:mn></mml:math></inline-formula>).

In this case, we write λ=|akl|1/(k+l) and (58)P(x,y)=λ-(k+l)akl(λx)k(λy)l+R(λxλ,λyλ)=Q(λx,λy). Therefore, (59)T+f(x)=p.v.eiQ(λx,λy)K(x-y)f(y)dy=p.v.eiQ(λx,y)K(λxλ-yλ)f(yλ)λ-1dy=Tλ+(f(·λ))(λx), where Kλ(x-y)=λ-1K(x/λ-y/λ) and (60)Tλ+f(x)=p.v.eiQ(x,y)Kλ(x-y)f(y)dy. It is easy to check that Kλ satisfies (2) and (10). We have thus established that (61)Tλ+fLp(w)CfLp(w) with similar statements as in Case 1. By Lemma 7, we have (62)|T+f(x)|pw(x)dx=|Tλ+f(·λ)(λx)|pw(x)dx=λ-1|Tλ+f(·λ)(x)|pw(xλ)dxC|f(xλ)|pw(xλ)dx=C|f(x)|pw(x)dx; that is, (63)T+fLp(w)CfLp(w), where C depends on the total degree of P(x,y) but not on the coefficients of P(x,y).

(2) We omit the details, since they are very similar to that of the proof of (1) with wAp- instead of wAp+.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work was partially supported by NSF of China (Grant nos. 11271175, 10931001, and 11301249), NSF of Shandong Province (Grant no. ZR2012AQ026), the AMEP and DYSP of Linyi University, and the Key Laboratory of Mathematics and Complex System (Beijing Normal University), Ministry of Education, China.

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